Fractal explanation of Meyer–Neldel rule

Journal of Non-Crystalline Solids 458 (2017) 137–140

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Fractal explanation of Meyer–Neldel rule Samy A. El-Sayed Phys. Dept., Faculty of Science, University of Beni-Suef, Beni-Suef, Egypt

a r t i c l e

i n f o

Article history: Received 10 November 2016 Received in revised form 14 December 2016 Accepted 19 December 2016 Available online 11 January 2017 PACS: 72.15.Cz 72.80.Ng 71.23.An 71.55.Jv

a b s t r a c t The relation between Meyer–Neldel rule (MNR) and fractal is investigated. Using very simple and elementary mathematics, it is proved that the (MNR) is one of Mandelbrot sets form. The dc conductivity in temperature range 1.7 K–300 K of structural disordered germanium irradiated fast neutrons with fluencies 1016 cm−2 ≤ Φ ≤ 1.2 × 1017 cm−2 is measured. From analysis, the dc conductivity results, the obtained EMN values is proportional with dc activation energies of each mechanisms of conduction. From the three values of EMN related to each mechanism of conduction the second stage of EMN is obtained. The second stage of EMN value is lower than EMN obtained from ΔE1 or ΔE2. This result confirms the smaller replicas of self-similar distribution of impurity centers in germanium disordered fast neutrons irradiation. EMN is postulated as the loss of energy dissipated in formation certain fractal system in a medium. © 2016 Elsevier B.V. All rights reserved.

1. Introduction In investigation of disorder materials, several properties, e.g., conductivity and diffusion are found to show exponential thermally activated (Arrhenius) behavior [1].
 −ΔE X ¼ X0 exp KT

ð1Þ

Here X is the absolute rate of a thermally activated process, X0 the preexponential factor, ΔE the activation energy and k the Boltzmann constant. Meyer and Neldel detected that X0 is linked to ΔE by the following relation:
X0 ¼ X00 exp

ΔE EMN

 ð2Þ

where X00 and EMN are positive constants. EMN is known as the Meyer– Neldel energy for the process in question. This pragmatic relation is known as the MN rule (MNR) or the compensation effect [2]. This relation known as the constable law which has been observed in many different materials for a number of phenomena [3–6]. The MNR is observed in numerous thermally activated phenomena e.g. kinetics (hopping) and thermodynamics, in crystalline, amorphous, and liquid semiconductors. Innumerable different models have been suggested to explain the MNR in different systems such as extrinsic

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semiconductors, amorphous and polycrystalline semiconductors. A model, called the statistical shift model, in which the Fermi level shifts with temperatures because of asymmetry between the density of states (DOS) above and below it, has been used to account for the MNR in electronic semiconductors. However, it could not answer the question of why some non-electronic conductors follow the MNR, where crystalline silicon does not? Also, Busch suggested that the MNR in extrinsic broad band semiconductors is due to a freezing of the donor concentration during cooling after the preparation [7]. The MNR in amorphous or polycrystalline semiconductors may derive from an exponential tailing of the majority band states as suggested by Roberts and by Cohen et al., or it may be due to a long-ranged electrostatic random potential. As regards the MNR in organic semiconductors, Kemeny and Rosenberg proposed a model where electrons or polaron tunnel through intermolecular barriers from activated energy states of the organic molecules. For ionic conductors it has been argued that the MNR is an approximate relation valid for ionic crystals with either Frenkel, Schottky or interstitial disorder. It has even been suggested that the MNR may be a spurious effect due to a thin rectifying layer at the electrode-solid interface. For the explanation of problems associated with MNR, Yelon and Movaghar [8] have proposed a model called as YM model. According to this model, the MNR arises for kinetic procedures in which ΔE is the energy of kinetic barrier and for which ΔE is large compared to the energies excitations which contribute to the activations as well as KT. Yelon et al. suggested the optical phonons are the source of excitation energy in such process. It is assumed that many phonons engage in trapping and detrapping of electrons either by cascade or by multi phonon process. They have described MNR with entropy term which may fluctuate the pre-factor by many orders of magnitude as it goes in equally well to crystalline or

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amorphous material, [9]. Also Fang [10] proved the equivalence of MNR with the Schewiedler relaxation distribution. “Clouds are not spheres, mountains are not cones, coastlines are not circles, bark not smooth, nor does lightning travel in a straight line” [11]. Fractal structures are found everywhere in nature [12]. The possible simplest definition of fractal is an object which appears self-similar under varying degrees of magnification [13]. Some disordered materials possess fractal structure over a certain range of length scales. Introduction of the fractal concept has been successful in the identification of many of the physical properties in disordered systems [14]. Static (geometric) properties of fractals have been widely accepted as describing properties of large collection physical objects that exhibit some kind of self-similarity [15]. Fractals may be found in nature or generated using a mathematical recipe [13]. The word fractal was coined by Benoit Mandelbrot, sometimes referred to as the father of fractal geometry. Mandelbrot realized that it is very often impossible to describe nature using only Euclidean geometry. Mandelbrot proposed that fractals and geometrical fractals could be used to describe real objects [16]. The term percolation connected with a new class of mathematics concerned with the flow of a liquid through a random maze [17]. Shklovskii and Efros on the basis of the percolation theory derived the following formula: σ ¼ σ 3 eE3 =KT where −Nα a

σ 3 ¼ σ 03 e

D

; α ¼ 1:74  0:02;

ð3Þ

and E3 is the hopping conduction energy. To describe the conductivity (σ) at hopping conduction region (at low temperatures), where ND is the donor concentration and (a) Bohr radius of impurity electron wave, i.e.:

Multiplying denominator and numerator of the exponent of Eq. (5) pffiffiffiffiffiffiffiffi −1

by i =

iΔE

ð−1Þx0 ¼ x00 eiEMN  eiðϑþπÞ iðΔE þπÞ= iEMN

−x0 ¼ x00 e

ΔE EMN ln θ ¼ ln ΔE þ i ln EMN −x0 ¼ x00 eiðϑþπÞ ; ð1−Þn x0 ¼ x00 einðϑþπÞ Let ϑ ¼ −i

ð6Þ

And by using Demoiver's theorem, it is easy to prove that: z ¼ ð−1Þ1−n zn

ð7Þ

where z is the complex number equation. It is obvious that Eq. (7) belong to Mandelbrot sets [10]: z ¼ z2 þ c

ð8Þ

Latt'es 1918, has pointed out that two collections of functions of complex variables share the property that the doubling formula yielding f (2z) is a rational function of f(z). On such collection centers, the function tanθ, tanhθ, and cosθ, involve only minor modifications. Historically the study of global properties of iterations began with Cayley 1879 used the Newton–Raphson method to solve the equation f (z) = z2 – 1 = 0. That method consists in iterations of rational function g (z) = (z2 – 1) / 2z = λ(z + 1 / z) [15]. Levinshtien et al. [20] suggested an approach based on an extension of the scaling hypothesis. They assumed that as we approach the percolation threshold the large-scale structure of the network remains similar to itself in in the sense that its topology remains the same and its linear dimension change proportionally with the correlating distances L = ∣τ∣−υ, where τ = x − xc. The critical exponents of Insulator-Metal transition in germanium disordered neutron irradiation was determined [19] and were found to be ≤2 as predicted by scaling theory. 3. Experimental

ln σ 03 ¼ σ 003 −

α 1=3

ND a

ð4Þ

Here we have to notice the similarity between Eq. (2) and Eq. (3), in that the two equations having exponent form. But Eq. (3) deduced using percolation theory. In case of random resistance networks the essential role played by randomness of current carrying properties. When randomness is strong as embodied by a network consisting of random mixture of resistors and insulators, this random resistor network undergoes transition between a conducting phase and insulating phase when the resistor concentration passes through a percolation threshold [18]. In Germanium doped irradiation defects the critical behavior of activation energy and also the parameters which define them, i.e. the dielectric constant, localization radius, and the coefficient To are found and compared with predictions of scaling theory [19]. Is there any relation between MNR and the fractal? 2. Theoretical Since Meyer–Neldel Rule has the form ΔE

x ¼ x0 e−KT ; ΔE x0 ¼ x00 eEMN eiπ ¼ −1; einπ ¼ ð−1Þn

But from Euler's formula

ð5Þ

Five samples of non-doped germanium with carrier density no = 3 × 013 cm−3 were irradiated with fast reactor neutrons with energies E ≥ 0.1 MeV in the range 6 × 1016 cm−2 ≤ Φ ≤ 1.2 × 1017 cm−2. As a result of irradiation all samples become disordered p-type [21,22]. In order to reduce the transmutation doping effect all samples were placed in 1 mm thick cadmium containers. During irradiation in the reactor the ratio between thermal neutrons fluency and the fluency of fast neutrons was about ten. So, it was possible to obtain samples of germanium “doped” with acceptor-like radiation defects {Ge (RD)}. To ensure that the electrical properties controlled by the transmutation doping, a complete annealing is performed at 450 °C for 24 h. For electrical conductivity measurements a special double wall glassy cryostat is designed. The cryostat is attached with vacuum pump has evacuation rate faster than the evaporation rate of He4 gas, thus the pressure inside the cryostat is decreases and cause decrease the temperature inside the cryostat. For electrical resistivity measurements, the conventional four prop method is used. The electrochemical deposition technique (cold method) is used for precipitating Ni electrode in the desired position on the samples. Thin Cu wires are fixed above the Ni electrodes using indium. The samples were in parallelepiped shape with length about 8–12 mm, thickness about 1–2 mm, width about 2–3 mm and the electrode apart about 3–4 mm. Resistivity of Ge (RD) was measured in the temperature range from 1.7 K up to 300 K. The temperature was determined with a semiconductor thermistor in the interval 77.4–4.2 K, from saturated vapor pressure of He4 in the interval 4.2–1.5 K. The voltage across samples always ≤1 V and the current across the sample decreases from μA to nA as the temperature lowered. The electrical properties of Ge (RD) are determined solely by acceptor-like radiation defects in spite of fairly

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139

high concentration impurities introduced by transmutation doping. The least square method fitting, using a computer program, is used to analysis the data extracted. 4. Experimental results and discussion Fig. 1 shows the dependence of conductivity on temperature. This figure shows that the conductivity increases as the irradiation fluency increases. The three regions of conductivity are apparent, satisfying the thermally activated conductivity relation [22–24]: E ¼E  −ðEA −E F þW1 Þ W2 F C KT σ ¼ σ 1 e− KT þ σ 2 e þ σ 3 e− KT

ð9Þ Fig. 2. Shows ln (σo) = f (ΔE) for the three mechanisms of electrical conductivity.

where the term (σ1 exp. − (EC − EF) / KT) represents transport by carriers excited beyond the mobility edges into non-localized (extended) states at EC or EV. And σ is the conductivity, σ1 is the pre-exponential factor. A plot of lnσ versus 1 / T will yield a straight line if (EC − EF) is a linear function of T over the temperature range measured, and the intersection of line extension with the ordinate gives the σ1. As the temperature decreases transport carriers excited into localized states at the band edges and the conductivity is given by σ2exp. − (EA − EF + W1) / KT, where W1 is the activation energy for carriers hopping, W1 should decrease with decreasing temperature on the account of variable - range nature of the hopping transport. However, as the temperature dependence is through the carrier activation term, an approximately linear dependence of lnσ versus 1 / T is again expected. As the temperature lowered more, there will be a contribution from carriers with energies near EF which can hop between localized states. This contribution is described by σ3 exp. (−W2) / KT, where σ1 ≥ σ2 and W is the hopping energy, of the order of half the width of the band of the states. The nominators of the exponentials of Eq. (9) are simply labeled in the present article by the respective symbols ΔE1, ΔE2 and ΔE3. Fig. 2 shows lnσi = f (ΔEi), i = 1: 3, for the three mechanisms of conductivity. It is clear that the slope is positive for all mechanisms of conductivity i.e. the applicability of (MNR) in germanium structural disordered fast reactor neutrons (Table 1).

The MNR fitting equations of the three mechanisms of dc conduction for the samples under investigations are

ð10Þ

By plotting the obtained values of (EMN)−1 of the three mechanisms of conduction versus the intersection with the ordinate as seen in Eq. (10), Fig. 3 is obtained from which the energy EMN value of second stage of MNR is calculated and equal to 0.0007 eV. Here it worth to notice that the EMN values of the first stage of MNR is decreases as the mechanism of conduction changes from Band to band conduction mechanism ΔE1, to hopping by carriers into localized states at the band edges of energies close to band edges ΔE2, and hoping of charge carriers with energies near Fermi level ΔE3. It is also noticed that the EMN is directly proportional with value of dc conductivity activation energy of each mechanism of conduction. Since ΔE1 N ΔE2 N ΔE3 the obtained EMN is also decreases with change the mechanism of conduction which is changes from mechanism to another as the temperature decreases [22]. The value of the second stage of EMN is lower than the values of EMN related to ΔE1 or ΔE2 But much higher than EMN related to ΔE3. This results confirm the postulation that Meyer– Neldel Rule (Eq. (2)) is one of the Mandelbrot's sets form. The most important property of fractals is their self- similarity, or their symmetry under dilation. Construction of fractal begins with initiator and proceeds with set of operations is called generator. The initiator is replaced by smaller replicas of itself and fractals build inwardly, towards ever smaller length scale [12]. Since ϑ ¼ −i EΔE , from Eq. (6) ΔE is a real value, one MN can interpret EMN as the energy dissipated in formation certain fractal system (certain impurity distribution in a certain medium). The decrease of experimentally obtained values EMN from stage one to stage two confirms the smaller replicas of self-similar distribution of impurity centers in germanium irradiated reactor neutrons. Using standard transient capacitance spectroscopy and C-V measurements Fourches [23,24] proved the existence of P-type regions with significant spatial extension in fast neutrons irradiated with high purity germanium. Also, Muramatsu et al. [25], used the fractal theory for explanation the medium range order of amorphous thin film samples. Table 1 The radiation fluency (φ), and the three activation energies (ΔE1, ΔE2 ,ΔE3) for the three mechanisms of conductivity.

Fig. 1. The dependence of dc conductivity on temperature.

Sample no.

φ, n.cm−2

ΔE1, eV

ΔE2, eV

1 2 3 4 5

6.00E 6.90E 9.00E 1.10E 1.20E

0.01 ± 1E−3 0.012 ± 1E−3 0.012 ± 1E−3 0.007 ± 1E−3 0.009 ± 1E−3

0.002 0.006 0.004 0.001 0.001

+ + + + +

16 16 16 17 17

± ± ± ± ±

ΔE3, eV 3E−4 3E−4 3E−4 3E−4 3E−4

0.001 ± 5E−4 0.00015 ± 5E−4 0.001 ± 5E−4 0.0006 ± 5E−4 0.0008 ± 5E−4

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for ΔE3. EMN is postulated as the loss of energy dissipated in formation certain fractal system in a medium. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] Fig. 3. Shows ln (ρoo) = f (1 / EMN) as obtained from the least square method applied on the data represented in Fig. 2.

[13]

5. Conclusions

[14] [15] [16]

The overall results show that; the Meyer–Neldel rule may be belong to Mandelbrot sets. The dc conductivity of germanium structural disordered fast reactor neutrons with fluencies 1016 cm−2 ≤ Φ ≤ 1.2 × 1017 cm−2 is measured is measured in the temperature range 1.7 K–300 K. The three activations energies ΔE1, ΔE2 and ΔE3 of the three conventional mechanisms of conductivity are obtained. By applying MNR the EMN of the three mechanisms of conductivity are obtained. From the deduced EMN and the intersection of the straight line with the ordinate the second stage EMN is obtained. The obtained the value of second stage EMN is lower than the EMN obtained for ΔE1 or ΔE2 but much higher than EMN

[17] [18] [19] [20] [21] [22] [23] [24] [25]

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